The first two conditions ensures the control and state trajectories are admissible, i.e., they do not violate any control or state constraints. In this work we consider box form control constraints. The third condition is a set of joint state-control path inequality constraints at each point along the horizon. The last condition is a set of terminal constraints, which differs from the horizon reward function \(\Phi (x(t_H))\) . In most cases the optimal control problem cannot be solved analytically. Rather, so-called direct transcription methods [1]}, [2]} or indirect methods [3]} can be applied to solve the problem numerically, in which case the dynamics (REF ) can be approximated with a numerical discretization of the form
\( x_{k+1} = F_k(x_k,u_k)\) 
